Population Game Model for Epidemic Dynamics with Two Classes of Vaccine-induced Immunity


  • P. Phang Department of Mathematics and Statistics, Curtin University, Perth, Australia. Department of Computational Science and Mathematics, Faculty of Computer Science and Information Technology, Universiti Malaysia Sarawak, 94300 Kota Samarahan, Sarawak, Malaysia.
  • B. Wiwatanapataphee Department of Mathematics and Statistics, Curtin University, Perth, Australia.
  • Y-H. Wu Department of Mathematics and Statistics, Curtin University, Perth, Australia.


Two Classes of Vaccine-Induced Immunity, Vaccination Population Games, Vaccine Efficacy,


Behavioural factors play a key and pivotal role in the success of a voluntary vaccination programme for combating infectious diseases. Individuals usually base their voluntary vaccination decisions on the perceived costs of vaccination and infection. The perceived cost of vaccination is easily influenced by the degree of protection conferred by vaccines against infection, also known as vaccine efficacy. Although certain vaccines have a decrease in its effectiveness in specific duration of time, they do offer a reduction of transmissibility and faster recovery for vaccinated infected individuals. These additional characteristics of imperfect vaccines are well-captured in an epidemic model with two classes of vaccine-induced immunity. In this paper, the interplays between these characteristics of vaccines, the dynamics of vaccination uptake and epidemics are investigated in the vaccination population games framework. Specifically, we study to what extent the population- and individual-level vaccination rates are influenced by these characteristics of vaccines at equilibrium state.


S. Funk, M. Salathé, and V.A. Jansen, “Modelling the influence of human behaviour on the spread of infectious diseases: a review,” J. of the Roy. Soc. Interface, vol. 7, no. 50, pp. 1247-1256, 2010.

G. A. Weinberg, and P. G. Szilagyi, “Vaccine epidemiology: efficacy, effectiveness, and the translational research roadmap,” J. of Infectious Diseases, vol. 201, no. 11, pp. 1607-1610, 2010.

S. Gandon, M. Mackinnon, S. Nee, and A. Read, “Imperfect vaccination: some epidemiological and evolutionary consequences,” Proc. Roy. Soc. of London B: Biological Sci., vol. 270, no. 1520, pp.1129-1136, 2003.

F. Magpantay, M. Riolo, M. D. de Cellès, A. A. King, and P. Rohani, “Epidemiological consequences of imperfect vaccines for immunizing infections,” SIAM J. on Appl. Math., vol. 74, no.6, pp. 1810-1830, 2014.

M. Keeling, M. Tildesley, T. House, and L. Danon, “The mathematics of vaccination,” Math. Today, vol. 49, pp. 40-43, 2013.

(CDC)a, “Chickenpox vaccine: What you need to know,” http://www.cdc.gov/vaccines/hcp/vis/vis-statements/varicella.html, 2013. Accessed: 2016-08-28.

J. Heffernan and M. J. Keeling, “Implications of vaccination and waning immunity,” Proc. Roy. Soc. of London B: Biological Sci., pages rspb-2009, 2009.

(CDC)b, “Long-term effectiveness of whooping cough vaccines,” http://www.cdc.gov/pertussis/pregnant/mom/vacc-effectiveness.html, 2016. Accessed: 2016-08-28.

E. Elbasha, C. Podder, and A. Gumel, “Analyzing the dynamics of an SIRS vaccination model with waning natural and vaccine-induced immunity,” Nonlinear Analysis: Real World Applications, vol. 12, no.5, pp. 2692-2705, 2011.

B. Wu, F. Fu, and L. Wang, “Imperfect vaccine aggravates the longstanding dilemma of voluntary vaccination,” PLoS ONE, vol.6, no.6: e20577, 2011.

C. Wells and C. Bauch, “The impact of personal experiences with infection and vaccination on behaviour-incidence dynamics of seasonal influenza,” Epidemics, vol. 4, no. 3, pp. 139-151, 2012.

R. Vardavas, R. Breban, and S. Blower, “A universal long-term flu vaccine may not prevent severe epidemics,” BMC Research Notes, vol. 3, no. 1, p. 92, 2010.

F. Salvarani, and G. Turinici, “Individual vaccination equilibrium for imperfect vaccine efficacy and limited persistence,” https://hal.archives-ouvertes.fr/hal-01302557v2, 2016.

A. Cardillo, C. Reyer-Suárez, F. Naranjo, and J. Gómez-Gardenes, “Evolutionary vaccination dilemma in complex networks,” Phys. Rev. E, vol. 88, no. 3: 032803, 2013.

T. C. Reluga, and A. P. Galvani, “A general approach for population games with application to vaccination,” Math. Biosci., vol. 230, no. 2, pp. 67-78, 2011.

G. Szabó, and G. Fath, “Evolutionary games on graphs,” Phys. Rep., vol. 446, no. 4, pp. 97-216, 2007.

E. Shim, J. J. Grefenstette, S. M. Albert, B. E. Cakouros, and D. S. Burke, “A game dynamic model for vaccine skeptics and vaccine believers: measles as an example,” J. of Theor. Biol., vol. 295, pp. 194-203, 2012.

P. Stephanie, “Vaccination and other altruistic medical treatments: should autonomy or communitarianism prevail?” Medical Law International, vol. 4, no. 3-4, pp. 223-243, 2000.

Y. Tsutsui, U. Benzion, and S. Shahrabani, “Economic and behavioural factors in an individual’s decision to take the influenza vaccination in Japan,” The J. of Socio-Economics, vol. 41, no. 5, pp. 594-602, 2012.




How to Cite

Phang, P., Wiwatanapataphee, B., & Wu, Y.-H. (2017). Population Game Model for Epidemic Dynamics with Two Classes of Vaccine-induced Immunity. Journal of Telecommunication, Electronic and Computer Engineering (JTEC), 9(2-9), 31–36. Retrieved from https://jtec.utem.edu.my/jtec/article/view/2673